Quasi-local conserved quantities in general relativity

In General Relativity, because of the equivalence principle, a local definition of energy does not exist. The hope has been that it will be possible to define energy quasi-locally. However, there exists no general framework in which a definition of quasi-local energy is sufficiently understood. In this thesis, an attempt is made to provide such a framework. In the first part of this thesis, we propose a general prescription for defining quasi-local conserved quantities in General Relativity. Our starting point is the construction of conserved quantities by Wald and Zoupas at null infinity. We point out why their construction is not applicable in the bulk of a spacetime and therefore cannot be used to define quasi-local conserved quantities. Then we propose a modification of their prescription so that the conserved quantities are defined more generally, and in particular in the bulk of a spacetime. We proceed by applying our construction to BMS symmetries. These are asymptotic symmetries of asymptotically flat spacetimes, which at null infinity are understood to define BMS charges including the Bondi mass. We discuss how to extend the notion of BMS symmetry into the bulk of the spacetime so as to define quasi-local conserved BMS charges there. We then argue that the zero mode of this charge is a promising definition of quasi-local energy. In the second part of the thesis, we study the gravitational memory effect, which is a statement about a permanent displacement between geodesics after the passing of a burst of radiation. At present, it is best understood on a flat background, where one has sufficient control over a notion of "static observer'', which is used to measure the displacement. However, on a curved background, such as a black hole, it is usually not clear how to quantify the gravitational memory effect. We shall propose a new method to detect gravitational memory. We show that the method is applicable on a black hole background. Furthermore, we make a connection between our formulation of the memory effect and BMS symmetries. This extends a previously discovered connection at null infinity to the bulk of the spacetime